Require Export fintype. Require Export header_extensible.
Section ty.
Inductive ty (nty : nat) : Type :=
| var_ty : (fin) (nty) -> ty (nty)
| top : ty (nty)
| arr : ty (nty) -> ty (nty)
| all : ty ((S) nty) -> ty (nty)
| recty : list (prod (nat ) (ty (nty))) -> ty (nty).
Lemma congr_top { mty : nat } : top (mty) = top (mty) .
Proof. congruence. Qed.
Lemma congr_arr { mty : nat } { s0 : ty (mty) } { t0 : ty (mty) } (H1 : s0 = t0) : arr (mty) s0 = arr (mty) t0 .
Proof. congruence. Qed.
Lemma congr_all { mty : nat } { s0 : ty ((S) mty) } { t0 : ty ((S) mty) } (H1 : s0 = t0) : all (mty) s0 = all (mty) t0 .
Proof. congruence. Qed.
Lemma congr_recty { mty : nat } { s0 : list (prod (nat ) (ty (mty))) } { t0 : list (prod (nat ) (ty (mty))) } (H1 : s0 = t0) : recty (mty) s0 = recty (mty) t0 .
Proof. congruence. Qed.
Definition upRen_ty_ty { m : nat } { n : nat } (xi : (fin) (m) -> (fin) (n)) : (fin) ((S) (m)) -> (fin) ((S) (n)) :=
(up_ren) xi.
Definition upRenList_ty_ty (p : nat) { m : nat } { n : nat } (xi : (fin) (m) -> (fin) (n)) : (fin) (p+ (m)) -> (fin) (p+ (n)) :=
upRen_p p xi.
Fixpoint ren_ty { mty : nat } { nty : nat } (xity : (fin) (mty) -> (fin) (nty)) (s : ty (mty)) : ty (nty) :=
match s with
| var_ty (_) s => (var_ty (nty)) (xity s)
| top (_) => top (nty)
| arr (_) s0 => arr (nty) ((ren_ty xity) s0)
| all (_) s0 => all (nty) ((ren_ty (upRen_ty_ty xity)) s0)
| recty (_) s0 => recty (nty) ((list_map (prod_map (fun x => x) (ren_ty xity))) s0)
end.
Definition up_ty_ty { m : nat } { nty : nat } (sigma : (fin) (m) -> ty (nty)) : (fin) ((S) (m)) -> ty ((S) nty) :=
(scons) ((var_ty ((S) nty)) (var_zero)) ((funcomp) (ren_ty (shift)) sigma).
Definition upList_ty_ty (p : nat) { m : nat } { nty : nat } (sigma : (fin) (m) -> ty (nty)) : (fin) (p+ (m)) -> ty (p+ nty) :=
scons_p p ((funcomp) (var_ty (p+ nty)) (zero_p p)) ((funcomp) (ren_ty (shift_p p)) sigma).
Fixpoint subst_ty { mty : nat } { nty : nat } (sigmaty : (fin) (mty) -> ty (nty)) (s : ty (mty)) : ty (nty) :=
match s with
| var_ty (_) s => sigmaty s
| top (_) => top (nty)
| arr (_) s0 => arr (nty) ((subst_ty sigmaty) s0)
| all (_) s0 => all (nty) ((subst_ty (up_ty_ty sigmaty)) s0)
| recty (_) s0 => recty (nty) ((list_map (prod_map (fun x => x) (subst_ty sigmaty))) s0)
end.
Definition upId_ty_ty { mty : nat } (sigma : (fin) (mty) -> ty (mty)) (Eq : forall x, sigma x = (var_ty (mty)) x) : forall x, (up_ty_ty sigma) x = (var_ty ((S) mty)) x :=
fun n => match n with
| Some fin_n => (ap) (ren_ty (shift)) (Eq fin_n)
| None => eq_refl
end.
Definition upIdList_ty_ty { p : nat } { mty : nat } (sigma : (fin) (mty) -> ty (mty)) (Eq : forall x, sigma x = (var_ty (mty)) x) : forall x, (upList_ty_ty p sigma) x = (var_ty (p+ mty)) x :=
fun n => scons_p_eta (var_ty (p+ mty)) (fun n => (ap) (ren_ty (shift_p p)) (Eq n)) (fun n => eq_refl).
Fixpoint idSubst_ty { mty : nat } (sigmaty : (fin) (mty) -> ty (mty)) (Eqty : forall x, sigmaty x = (var_ty (mty)) x) (s : ty (mty)) : subst_ty sigmaty s = s :=
match s with
| var_ty (_) s => Eqty s
| top (_) => congr_top
| arr (_) s0 => congr_arr ((idSubst_ty sigmaty Eqty) s0)
| all (_) s0 => congr_all ((idSubst_ty (up_ty_ty sigmaty) (upId_ty_ty (_) Eqty)) s0)
| recty (_) s0 => congr_recty ((list_id (prod_id (fun x => (eq_refl) x) (idSubst_ty sigmaty Eqty))) s0)
end.
Definition upExtRen_ty_ty { m : nat } { n : nat } (xi : (fin) (m) -> (fin) (n)) (zeta : (fin) (m) -> (fin) (n)) (Eq : forall x, xi x = zeta x) : forall x, (upRen_ty_ty xi) x = (upRen_ty_ty zeta) x :=
fun n => match n with
| Some fin_n => (ap) (shift) (Eq fin_n)
| None => eq_refl
end.
Definition upExtRen_list_ty_ty { p : nat } { m : nat } { n : nat } (xi : (fin) (m) -> (fin) (n)) (zeta : (fin) (m) -> (fin) (n)) (Eq : forall x, xi x = zeta x) : forall x, (upRenList_ty_ty p xi) x = (upRenList_ty_ty p zeta) x :=
fun n => scons_p_congr (fun n => eq_refl) (fun n => (ap) (shift_p p) (Eq n)).
Fixpoint extRen_ty { mty : nat } { nty : nat } (xity : (fin) (mty) -> (fin) (nty)) (zetaty : (fin) (mty) -> (fin) (nty)) (Eqty : forall x, xity x = zetaty x) (s : ty (mty)) : ren_ty xity s = ren_ty zetaty s :=
match s with
| var_ty (_) s => (ap) (var_ty (nty)) (Eqty s)
| top (_) => congr_top
| arr (_) s0 => congr_arr ((extRen_ty xity zetaty Eqty) s0)
| all (_) s0 => congr_all ((extRen_ty (upRen_ty_ty xity) (upRen_ty_ty zetaty) (upExtRen_ty_ty (_) (_) Eqty)) s0)
| recty (_) s0 => congr_recty ((list_ext (prod_ext (fun x => (eq_refl) x) (extRen_ty xity zetaty Eqty))) s0)
end.
Definition upExt_ty_ty { m : nat } { nty : nat } (sigma : (fin) (m) -> ty (nty)) (tau : (fin) (m) -> ty (nty)) (Eq : forall x, sigma x = tau x) : forall x, (up_ty_ty sigma) x = (up_ty_ty tau) x :=
fun n => match n with
| Some fin_n => (ap) (ren_ty (shift)) (Eq fin_n)
| None => eq_refl
end.
Definition upExt_list_ty_ty { p : nat } { m : nat } { nty : nat } (sigma : (fin) (m) -> ty (nty)) (tau : (fin) (m) -> ty (nty)) (Eq : forall x, sigma x = tau x) : forall x, (upList_ty_ty p sigma) x = (upList_ty_ty p tau) x :=
fun n => scons_p_congr (fun n => eq_refl) (fun n => (ap) (ren_ty (shift_p p)) (Eq n)).
Fixpoint ext_ty { mty : nat } { nty : nat } (sigmaty : (fin) (mty) -> ty (nty)) (tauty : (fin) (mty) -> ty (nty)) (Eqty : forall x, sigmaty x = tauty x) (s : ty (mty)) : subst_ty sigmaty s = subst_ty tauty s :=
match s with
| var_ty (_) s => Eqty s
| top (_) => congr_top
| arr (_) s0 => congr_arr ((ext_ty sigmaty tauty Eqty) s0)
| all (_) s0 => congr_all ((ext_ty (up_ty_ty sigmaty) (up_ty_ty tauty) (upExt_ty_ty (_) (_) Eqty)) s0)
| recty (_) s0 => congr_recty ((list_ext (prod_ext (fun x => (eq_refl) x) (ext_ty sigmaty tauty Eqty))) s0)
end.
Definition up_ren_ren_ty_ty { k : nat } { l : nat } { m : nat } (xi : (fin) (k) -> (fin) (l)) (tau : (fin) (l) -> (fin) (m)) (theta : (fin) (k) -> (fin) (m)) (Eq : forall x, ((funcomp) tau xi) x = theta x) : forall x, ((funcomp) (upRen_ty_ty tau) (upRen_ty_ty xi)) x = (upRen_ty_ty theta) x :=
up_ren_ren xi tau theta Eq.
Definition up_ren_ren_list_ty_ty { p : nat } { k : nat } { l : nat } { m : nat } (xi : (fin) (k) -> (fin) (l)) (tau : (fin) (l) -> (fin) (m)) (theta : (fin) (k) -> (fin) (m)) (Eq : forall x, ((funcomp) tau xi) x = theta x) : forall x, ((funcomp) (upRenList_ty_ty p tau) (upRenList_ty_ty p xi)) x = (upRenList_ty_ty p theta) x :=
up_ren_ren_p Eq.
Fixpoint compRenRen_ty { kty : nat } { lty : nat } { mty : nat } (xity : (fin) (mty) -> (fin) (kty)) (zetaty : (fin) (kty) -> (fin) (lty)) (rhoty : (fin) (mty) -> (fin) (lty)) (Eqty : forall x, ((funcomp) zetaty xity) x = rhoty x) (s : ty (mty)) : ren_ty zetaty (ren_ty xity s) = ren_ty rhoty s :=
match s with
| var_ty (_) s => (ap) (var_ty (lty)) (Eqty s)
| top (_) => congr_top
| arr (_) s0 => congr_arr ((compRenRen_ty xity zetaty rhoty Eqty) s0)
| all (_) s0 => congr_all ((compRenRen_ty (upRen_ty_ty xity) (upRen_ty_ty zetaty) (upRen_ty_ty rhoty) (up_ren_ren (_) (_) (_) Eqty)) s0)
| recty (_) s0 => congr_recty ((list_comp (prod_comp (fun x => (eq_refl) x) (compRenRen_ty xity zetaty rhoty Eqty))) s0)
end.
Definition up_ren_subst_ty_ty { k : nat } { l : nat } { mty : nat } (xi : (fin) (k) -> (fin) (l)) (tau : (fin) (l) -> ty (mty)) (theta : (fin) (k) -> ty (mty)) (Eq : forall x, ((funcomp) tau xi) x = theta x) : forall x, ((funcomp) (up_ty_ty tau) (upRen_ty_ty xi)) x = (up_ty_ty theta) x :=
fun n => match n with
| Some fin_n => (ap) (ren_ty (shift)) (Eq fin_n)
| None => eq_refl
end.
Definition up_ren_subst_list_ty_ty { p : nat } { k : nat } { l : nat } { mty : nat } (xi : (fin) (k) -> (fin) (l)) (tau : (fin) (l) -> ty (mty)) (theta : (fin) (k) -> ty (mty)) (Eq : forall x, ((funcomp) tau xi) x = theta x) : forall x, ((funcomp) (upList_ty_ty p tau) (upRenList_ty_ty p xi)) x = (upList_ty_ty p theta) x :=
fun n => (eq_trans) (scons_p_comp' (_) (_) (_) n) (scons_p_congr (fun z => scons_p_head' (_) (_) z) (fun z => (eq_trans) (scons_p_tail' (_) (_) (xi z)) ((ap) (ren_ty (shift_p p)) (Eq z)))).
Fixpoint compRenSubst_ty { kty : nat } { lty : nat } { mty : nat } (xity : (fin) (mty) -> (fin) (kty)) (tauty : (fin) (kty) -> ty (lty)) (thetaty : (fin) (mty) -> ty (lty)) (Eqty : forall x, ((funcomp) tauty xity) x = thetaty x) (s : ty (mty)) : subst_ty tauty (ren_ty xity s) = subst_ty thetaty s :=
match s with
| var_ty (_) s => Eqty s
| top (_) => congr_top
| arr (_) s0 => congr_arr ((compRenSubst_ty xity tauty thetaty Eqty) s0)
| all (_) s0 => congr_all ((compRenSubst_ty (upRen_ty_ty xity) (up_ty_ty tauty) (up_ty_ty thetaty) (up_ren_subst_ty_ty (_) (_) (_) Eqty)) s0)
| recty (_) s0 => congr_recty ((list_comp (prod_comp (fun x => (eq_refl) x) (compRenSubst_ty xity tauty thetaty Eqty))) s0)
end.
Definition up_subst_ren_ty_ty { k : nat } { lty : nat } { mty : nat } (sigma : (fin) (k) -> ty (lty)) (zetaty : (fin) (lty) -> (fin) (mty)) (theta : (fin) (k) -> ty (mty)) (Eq : forall x, ((funcomp) (ren_ty zetaty) sigma) x = theta x) : forall x, ((funcomp) (ren_ty (upRen_ty_ty zetaty)) (up_ty_ty sigma)) x = (up_ty_ty theta) x :=
fun n => match n with
| Some fin_n => (eq_trans) (compRenRen_ty (shift) (upRen_ty_ty zetaty) ((funcomp) (shift) zetaty) (fun x => eq_refl) (sigma fin_n)) ((eq_trans) ((eq_sym) (compRenRen_ty zetaty (shift) ((funcomp) (shift) zetaty) (fun x => eq_refl) (sigma fin_n))) ((ap) (ren_ty (shift)) (Eq fin_n)))
| None => eq_refl
end.
Definition up_subst_ren_list_ty_ty { p : nat } { k : nat } { lty : nat } { mty : nat } (sigma : (fin) (k) -> ty (lty)) (zetaty : (fin) (lty) -> (fin) (mty)) (theta : (fin) (k) -> ty (mty)) (Eq : forall x, ((funcomp) (ren_ty zetaty) sigma) x = theta x) : forall x, ((funcomp) (ren_ty (upRenList_ty_ty p zetaty)) (upList_ty_ty p sigma)) x = (upList_ty_ty p theta) x :=
fun n => (eq_trans) (scons_p_comp' (_) (_) (_) n) (scons_p_congr (fun x => (ap) (var_ty (p+ mty)) (scons_p_head' (_) (_) x)) (fun n => (eq_trans) (compRenRen_ty (shift_p p) (upRenList_ty_ty p zetaty) ((funcomp) (shift_p p) zetaty) (fun x => scons_p_tail' (_) (_) x) (sigma n)) ((eq_trans) ((eq_sym) (compRenRen_ty zetaty (shift_p p) ((funcomp) (shift_p p) zetaty) (fun x => eq_refl) (sigma n))) ((ap) (ren_ty (shift_p p)) (Eq n))))).
Fixpoint compSubstRen_ty { kty : nat } { lty : nat } { mty : nat } (sigmaty : (fin) (mty) -> ty (kty)) (zetaty : (fin) (kty) -> (fin) (lty)) (thetaty : (fin) (mty) -> ty (lty)) (Eqty : forall x, ((funcomp) (ren_ty zetaty) sigmaty) x = thetaty x) (s : ty (mty)) : ren_ty zetaty (subst_ty sigmaty s) = subst_ty thetaty s :=
match s with
| var_ty (_) s => Eqty s
| top (_) => congr_top
| arr (_) s0 => congr_arr ((compSubstRen_ty sigmaty zetaty thetaty Eqty) s0)
| all (_) s0 => congr_all ((compSubstRen_ty (up_ty_ty sigmaty) (upRen_ty_ty zetaty) (up_ty_ty thetaty) (up_subst_ren_ty_ty (_) (_) (_) Eqty)) s0)
| recty (_) s0 => congr_recty ((list_comp (prod_comp (fun x => (eq_refl) x) (compSubstRen_ty sigmaty zetaty thetaty Eqty))) s0)
end.
Definition up_subst_subst_ty_ty { k : nat } { lty : nat } { mty : nat } (sigma : (fin) (k) -> ty (lty)) (tauty : (fin) (lty) -> ty (mty)) (theta : (fin) (k) -> ty (mty)) (Eq : forall x, ((funcomp) (subst_ty tauty) sigma) x = theta x) : forall x, ((funcomp) (subst_ty (up_ty_ty tauty)) (up_ty_ty sigma)) x = (up_ty_ty theta) x :=
fun n => match n with
| Some fin_n => (eq_trans) (compRenSubst_ty (shift) (up_ty_ty tauty) ((funcomp) (up_ty_ty tauty) (shift)) (fun x => eq_refl) (sigma fin_n)) ((eq_trans) ((eq_sym) (compSubstRen_ty tauty (shift) ((funcomp) (ren_ty (shift)) tauty) (fun x => eq_refl) (sigma fin_n))) ((ap) (ren_ty (shift)) (Eq fin_n)))
| None => eq_refl
end.
Definition up_subst_subst_list_ty_ty { p : nat } { k : nat } { lty : nat } { mty : nat } (sigma : (fin) (k) -> ty (lty)) (tauty : (fin) (lty) -> ty (mty)) (theta : (fin) (k) -> ty (mty)) (Eq : forall x, ((funcomp) (subst_ty tauty) sigma) x = theta x) : forall x, ((funcomp) (subst_ty (upList_ty_ty p tauty)) (upList_ty_ty p sigma)) x = (upList_ty_ty p theta) x :=
fun n => (eq_trans) (scons_p_comp' ((funcomp) (var_ty (p+ lty)) (zero_p p)) (_) (_) n) (scons_p_congr (fun x => scons_p_head' (_) (fun z => ren_ty (shift_p p) (_)) x) (fun n => (eq_trans) (compRenSubst_ty (shift_p p) (upList_ty_ty p tauty) ((funcomp) (upList_ty_ty p tauty) (shift_p p)) (fun x => eq_refl) (sigma n)) ((eq_trans) ((eq_sym) (compSubstRen_ty tauty (shift_p p) (_) (fun x => (eq_sym) (scons_p_tail' (_) (_) x)) (sigma n))) ((ap) (ren_ty (shift_p p)) (Eq n))))).
Fixpoint compSubstSubst_ty { kty : nat } { lty : nat } { mty : nat } (sigmaty : (fin) (mty) -> ty (kty)) (tauty : (fin) (kty) -> ty (lty)) (thetaty : (fin) (mty) -> ty (lty)) (Eqty : forall x, ((funcomp) (subst_ty tauty) sigmaty) x = thetaty x) (s : ty (mty)) : subst_ty tauty (subst_ty sigmaty s) = subst_ty thetaty s :=
match s with
| var_ty (_) s => Eqty s
| top (_) => congr_top
| arr (_) s0 => congr_arr ((compSubstSubst_ty sigmaty tauty thetaty Eqty) s0)
| all (_) s0 => congr_all ((compSubstSubst_ty (up_ty_ty sigmaty) (up_ty_ty tauty) (up_ty_ty thetaty) (up_subst_subst_ty_ty (_) (_) (_) Eqty)) s0)
| recty (_) s0 => congr_recty ((list_comp (prod_comp (fun x => (eq_refl) x) (compSubstSubst_ty sigmaty tauty thetaty Eqty))) s0)
end.
Definition rinstInst_up_ty_ty { m : nat } { nty : nat } (xi : (fin) (m) -> (fin) (nty)) (sigma : (fin) (m) -> ty (nty)) (Eq : forall x, ((funcomp) (var_ty (nty)) xi) x = sigma x) : forall x, ((funcomp) (var_ty ((S) nty)) (upRen_ty_ty xi)) x = (up_ty_ty sigma) x :=
fun n => match n with
| Some fin_n => (ap) (ren_ty (shift)) (Eq fin_n)
| None => eq_refl
end.
Definition rinstInst_up_list_ty_ty { p : nat } { m : nat } { nty : nat } (xi : (fin) (m) -> (fin) (nty)) (sigma : (fin) (m) -> ty (nty)) (Eq : forall x, ((funcomp) (var_ty (nty)) xi) x = sigma x) : forall x, ((funcomp) (var_ty (p+ nty)) (upRenList_ty_ty p xi)) x = (upList_ty_ty p sigma) x :=
fun n => (eq_trans) (scons_p_comp' (_) (_) (var_ty (p+ nty)) n) (scons_p_congr (fun z => eq_refl) (fun n => (ap) (ren_ty (shift_p p)) (Eq n))).
Fixpoint rinst_inst_ty { mty : nat } { nty : nat } (xity : (fin) (mty) -> (fin) (nty)) (sigmaty : (fin) (mty) -> ty (nty)) (Eqty : forall x, ((funcomp) (var_ty (nty)) xity) x = sigmaty x) (s : ty (mty)) : ren_ty xity s = subst_ty sigmaty s :=
match s with
| var_ty (_) s => Eqty s
| top (_) => congr_top
| arr (_) s0 => congr_arr ((rinst_inst_ty xity sigmaty Eqty) s0)
| all (_) s0 => congr_all ((rinst_inst_ty (upRen_ty_ty xity) (up_ty_ty sigmaty) (rinstInst_up_ty_ty (_) (_) Eqty)) s0)
| recty (_) s0 => congr_recty ((list_ext (prod_ext (fun x => (eq_refl) x) (rinst_inst_ty xity sigmaty Eqty))) s0)
end.
Lemma rinstInst_ty { mty : nat } { nty : nat } (xity : (fin) (mty) -> (fin) (nty)) : ren_ty xity = subst_ty ((funcomp) (var_ty (nty)) xity) .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun x => rinst_inst_ty xity (_) (fun n => eq_refl) x)). Qed.
Lemma instId_ty { mty : nat } : subst_ty (var_ty (mty)) = id .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun x => idSubst_ty (var_ty (mty)) (fun n => eq_refl) ((id) x))). Qed.
Lemma rinstId_ty { mty : nat } : @ren_ty (mty) (mty) (id) = id .
Proof. exact ((eq_trans) (rinstInst_ty ((id) (_))) instId_ty). Qed.
Lemma varL_ty { mty : nat } { nty : nat } (sigmaty : (fin) (mty) -> ty (nty)) : (funcomp) (subst_ty sigmaty) (var_ty (mty)) = sigmaty .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun x => eq_refl)). Qed.
Lemma varLRen_ty { mty : nat } { nty : nat } (xity : (fin) (mty) -> (fin) (nty)) : (funcomp) (ren_ty xity) (var_ty (mty)) = (funcomp) (var_ty (nty)) xity .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun x => eq_refl)). Qed.
Lemma compComp_ty { kty : nat } { lty : nat } { mty : nat } (sigmaty : (fin) (mty) -> ty (kty)) (tauty : (fin) (kty) -> ty (lty)) (s : ty (mty)) : subst_ty tauty (subst_ty sigmaty s) = subst_ty ((funcomp) (subst_ty tauty) sigmaty) s .
Proof. exact (compSubstSubst_ty sigmaty tauty (_) (fun n => eq_refl) s). Qed.
Lemma compComp'_ty { kty : nat } { lty : nat } { mty : nat } (sigmaty : (fin) (mty) -> ty (kty)) (tauty : (fin) (kty) -> ty (lty)) : (funcomp) (subst_ty tauty) (subst_ty sigmaty) = subst_ty ((funcomp) (subst_ty tauty) sigmaty) .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun n => compComp_ty sigmaty tauty n)). Qed.
Lemma compRen_ty { kty : nat } { lty : nat } { mty : nat } (sigmaty : (fin) (mty) -> ty (kty)) (zetaty : (fin) (kty) -> (fin) (lty)) (s : ty (mty)) : ren_ty zetaty (subst_ty sigmaty s) = subst_ty ((funcomp) (ren_ty zetaty) sigmaty) s .
Proof. exact (compSubstRen_ty sigmaty zetaty (_) (fun n => eq_refl) s). Qed.
Lemma compRen'_ty { kty : nat } { lty : nat } { mty : nat } (sigmaty : (fin) (mty) -> ty (kty)) (zetaty : (fin) (kty) -> (fin) (lty)) : (funcomp) (ren_ty zetaty) (subst_ty sigmaty) = subst_ty ((funcomp) (ren_ty zetaty) sigmaty) .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun n => compRen_ty sigmaty zetaty n)). Qed.
Lemma renComp_ty { kty : nat } { lty : nat } { mty : nat } (xity : (fin) (mty) -> (fin) (kty)) (tauty : (fin) (kty) -> ty (lty)) (s : ty (mty)) : subst_ty tauty (ren_ty xity s) = subst_ty ((funcomp) tauty xity) s .
Proof. exact (compRenSubst_ty xity tauty (_) (fun n => eq_refl) s). Qed.
Lemma renComp'_ty { kty : nat } { lty : nat } { mty : nat } (xity : (fin) (mty) -> (fin) (kty)) (tauty : (fin) (kty) -> ty (lty)) : (funcomp) (subst_ty tauty) (ren_ty xity) = subst_ty ((funcomp) tauty xity) .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun n => renComp_ty xity tauty n)). Qed.
Lemma renRen_ty { kty : nat } { lty : nat } { mty : nat } (xity : (fin) (mty) -> (fin) (kty)) (zetaty : (fin) (kty) -> (fin) (lty)) (s : ty (mty)) : ren_ty zetaty (ren_ty xity s) = ren_ty ((funcomp) zetaty xity) s .
Proof. exact (compRenRen_ty xity zetaty (_) (fun n => eq_refl) s). Qed.
Lemma renRen'_ty { kty : nat } { lty : nat } { mty : nat } (xity : (fin) (mty) -> (fin) (kty)) (zetaty : (fin) (kty) -> (fin) (lty)) : (funcomp) (ren_ty zetaty) (ren_ty xity) = ren_ty ((funcomp) zetaty xity) .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun n => renRen_ty xity zetaty n)). Qed.
End ty.
Arguments var_ty {nty}.
Arguments top {nty}.
Arguments arr {nty}.
Arguments all {nty}.
Arguments recty {nty}.
Global Instance Subst_ty { mty : nat } { nty : nat } : Subst1 ((fin) (mty) -> ty (nty)) (ty (mty)) (ty (nty)) := @subst_ty (mty) (nty) .
Global Instance Ren_ty { mty : nat } { nty : nat } : Ren1 ((fin) (mty) -> (fin) (nty)) (ty (mty)) (ty (nty)) := @ren_ty (mty) (nty) .
Global Instance VarInstance_ty { mty : nat } : Var ((fin) (mty)) (ty (mty)) := @var_ty (mty) .
Notation "x '__ty'" := (var_ty x) (at level 5, format "x __ty") : subst_scope.
Notation "x '__ty'" := (@ids (_) (_) VarInstance_ty x) (at level 5, only printing, format "x __ty") : subst_scope.
Notation "'var'" := (var_ty) (only printing, at level 1) : subst_scope.
Class Up_ty X Y := up_ty : X -> Y.
Notation "↑__ty" := (up_ty) (only printing) : subst_scope.
Notation "↑__ty" := (up_ty_ty) (only printing) : subst_scope.
Global Instance Up_ty_ty { m : nat } { nty : nat } : Up_ty (_) (_) := @up_ty_ty (m) (nty) .
Notation "s [ sigmaty ]" := (subst_ty sigmaty s) (at level 7, left associativity, only printing) : subst_scope.
Notation "[ sigmaty ]" := (subst_ty sigmaty) (at level 1, left associativity, only printing) : fscope.
Notation "s ⟨ xity ⟩" := (ren_ty xity s) (at level 7, left associativity, only printing) : subst_scope.
Notation "⟨ xity ⟩" := (ren_ty xity) (at level 1, left associativity, only printing) : fscope.
Ltac auto_unfold := repeat unfold subst1, subst2, Subst1, Subst2, ids, ren1, ren2, Ren1, Ren2, Subst_ty, Ren_ty, VarInstance_ty.
Tactic Notation "auto_unfold" "in" "*" := repeat unfold subst1, subst2, Subst1, Subst2, ids, ren1, ren2, Ren1, Ren2, Subst_ty, Ren_ty, VarInstance_ty in *.
Ltac asimpl' := repeat first [progress rewrite ?instId_ty| progress rewrite ?compComp_ty| progress rewrite ?compComp'_ty| progress rewrite ?rinstId_ty| progress rewrite ?compRen_ty| progress rewrite ?compRen'_ty| progress rewrite ?renComp_ty| progress rewrite ?renComp'_ty| progress rewrite ?renRen_ty| progress rewrite ?renRen'_ty| progress rewrite ?varL_ty| progress rewrite ?varLRen_ty| progress (unfold up_ren, upRen_ty_ty, upRenList_ty_ty, up_ty_ty, upList_ty_ty)| progress (cbn [subst_ty ren_ty])| fsimpl].
Ltac asimpl := repeat try unfold_funcomp; auto_unfold in *; asimpl'; repeat try unfold_funcomp.
Tactic Notation "asimpl" "in" hyp(J) := revert J; asimpl; intros J.
Tactic Notation "auto_case" := auto_case (asimpl; cbn; eauto).
Tactic Notation "asimpl" "in" "*" := auto_unfold in *; repeat first [progress rewrite ?instId_ty in *| progress rewrite ?compComp_ty in *| progress rewrite ?compComp'_ty in *| progress rewrite ?rinstId_ty in *| progress rewrite ?compRen_ty in *| progress rewrite ?compRen'_ty in *| progress rewrite ?renComp_ty in *| progress rewrite ?renComp'_ty in *| progress rewrite ?renRen_ty in *| progress rewrite ?renRen'_ty in *| progress rewrite ?varL_ty in *| progress rewrite ?varLRen_ty in *| progress (unfold up_ren, upRen_ty_ty, upRenList_ty_ty, up_ty_ty, upList_ty_ty in *)| progress (cbn [subst_ty ren_ty] in *)| fsimpl in *].
Ltac substify := auto_unfold; try repeat (erewrite rinstInst_ty).
Ltac renamify := auto_unfold; try repeat (erewrite <- rinstInst_ty).
Section ty.
Inductive ty (nty : nat) : Type :=
| var_ty : (fin) (nty) -> ty (nty)
| top : ty (nty)
| arr : ty (nty) -> ty (nty)
| all : ty ((S) nty) -> ty (nty)
| recty : list (prod (nat ) (ty (nty))) -> ty (nty).
Lemma congr_top { mty : nat } : top (mty) = top (mty) .
Proof. congruence. Qed.
Lemma congr_arr { mty : nat } { s0 : ty (mty) } { t0 : ty (mty) } (H1 : s0 = t0) : arr (mty) s0 = arr (mty) t0 .
Proof. congruence. Qed.
Lemma congr_all { mty : nat } { s0 : ty ((S) mty) } { t0 : ty ((S) mty) } (H1 : s0 = t0) : all (mty) s0 = all (mty) t0 .
Proof. congruence. Qed.
Lemma congr_recty { mty : nat } { s0 : list (prod (nat ) (ty (mty))) } { t0 : list (prod (nat ) (ty (mty))) } (H1 : s0 = t0) : recty (mty) s0 = recty (mty) t0 .
Proof. congruence. Qed.
Definition upRen_ty_ty { m : nat } { n : nat } (xi : (fin) (m) -> (fin) (n)) : (fin) ((S) (m)) -> (fin) ((S) (n)) :=
(up_ren) xi.
Definition upRenList_ty_ty (p : nat) { m : nat } { n : nat } (xi : (fin) (m) -> (fin) (n)) : (fin) (p+ (m)) -> (fin) (p+ (n)) :=
upRen_p p xi.
Fixpoint ren_ty { mty : nat } { nty : nat } (xity : (fin) (mty) -> (fin) (nty)) (s : ty (mty)) : ty (nty) :=
match s with
| var_ty (_) s => (var_ty (nty)) (xity s)
| top (_) => top (nty)
| arr (_) s0 => arr (nty) ((ren_ty xity) s0)
| all (_) s0 => all (nty) ((ren_ty (upRen_ty_ty xity)) s0)
| recty (_) s0 => recty (nty) ((list_map (prod_map (fun x => x) (ren_ty xity))) s0)
end.
Definition up_ty_ty { m : nat } { nty : nat } (sigma : (fin) (m) -> ty (nty)) : (fin) ((S) (m)) -> ty ((S) nty) :=
(scons) ((var_ty ((S) nty)) (var_zero)) ((funcomp) (ren_ty (shift)) sigma).
Definition upList_ty_ty (p : nat) { m : nat } { nty : nat } (sigma : (fin) (m) -> ty (nty)) : (fin) (p+ (m)) -> ty (p+ nty) :=
scons_p p ((funcomp) (var_ty (p+ nty)) (zero_p p)) ((funcomp) (ren_ty (shift_p p)) sigma).
Fixpoint subst_ty { mty : nat } { nty : nat } (sigmaty : (fin) (mty) -> ty (nty)) (s : ty (mty)) : ty (nty) :=
match s with
| var_ty (_) s => sigmaty s
| top (_) => top (nty)
| arr (_) s0 => arr (nty) ((subst_ty sigmaty) s0)
| all (_) s0 => all (nty) ((subst_ty (up_ty_ty sigmaty)) s0)
| recty (_) s0 => recty (nty) ((list_map (prod_map (fun x => x) (subst_ty sigmaty))) s0)
end.
Definition upId_ty_ty { mty : nat } (sigma : (fin) (mty) -> ty (mty)) (Eq : forall x, sigma x = (var_ty (mty)) x) : forall x, (up_ty_ty sigma) x = (var_ty ((S) mty)) x :=
fun n => match n with
| Some fin_n => (ap) (ren_ty (shift)) (Eq fin_n)
| None => eq_refl
end.
Definition upIdList_ty_ty { p : nat } { mty : nat } (sigma : (fin) (mty) -> ty (mty)) (Eq : forall x, sigma x = (var_ty (mty)) x) : forall x, (upList_ty_ty p sigma) x = (var_ty (p+ mty)) x :=
fun n => scons_p_eta (var_ty (p+ mty)) (fun n => (ap) (ren_ty (shift_p p)) (Eq n)) (fun n => eq_refl).
Fixpoint idSubst_ty { mty : nat } (sigmaty : (fin) (mty) -> ty (mty)) (Eqty : forall x, sigmaty x = (var_ty (mty)) x) (s : ty (mty)) : subst_ty sigmaty s = s :=
match s with
| var_ty (_) s => Eqty s
| top (_) => congr_top
| arr (_) s0 => congr_arr ((idSubst_ty sigmaty Eqty) s0)
| all (_) s0 => congr_all ((idSubst_ty (up_ty_ty sigmaty) (upId_ty_ty (_) Eqty)) s0)
| recty (_) s0 => congr_recty ((list_id (prod_id (fun x => (eq_refl) x) (idSubst_ty sigmaty Eqty))) s0)
end.
Definition upExtRen_ty_ty { m : nat } { n : nat } (xi : (fin) (m) -> (fin) (n)) (zeta : (fin) (m) -> (fin) (n)) (Eq : forall x, xi x = zeta x) : forall x, (upRen_ty_ty xi) x = (upRen_ty_ty zeta) x :=
fun n => match n with
| Some fin_n => (ap) (shift) (Eq fin_n)
| None => eq_refl
end.
Definition upExtRen_list_ty_ty { p : nat } { m : nat } { n : nat } (xi : (fin) (m) -> (fin) (n)) (zeta : (fin) (m) -> (fin) (n)) (Eq : forall x, xi x = zeta x) : forall x, (upRenList_ty_ty p xi) x = (upRenList_ty_ty p zeta) x :=
fun n => scons_p_congr (fun n => eq_refl) (fun n => (ap) (shift_p p) (Eq n)).
Fixpoint extRen_ty { mty : nat } { nty : nat } (xity : (fin) (mty) -> (fin) (nty)) (zetaty : (fin) (mty) -> (fin) (nty)) (Eqty : forall x, xity x = zetaty x) (s : ty (mty)) : ren_ty xity s = ren_ty zetaty s :=
match s with
| var_ty (_) s => (ap) (var_ty (nty)) (Eqty s)
| top (_) => congr_top
| arr (_) s0 => congr_arr ((extRen_ty xity zetaty Eqty) s0)
| all (_) s0 => congr_all ((extRen_ty (upRen_ty_ty xity) (upRen_ty_ty zetaty) (upExtRen_ty_ty (_) (_) Eqty)) s0)
| recty (_) s0 => congr_recty ((list_ext (prod_ext (fun x => (eq_refl) x) (extRen_ty xity zetaty Eqty))) s0)
end.
Definition upExt_ty_ty { m : nat } { nty : nat } (sigma : (fin) (m) -> ty (nty)) (tau : (fin) (m) -> ty (nty)) (Eq : forall x, sigma x = tau x) : forall x, (up_ty_ty sigma) x = (up_ty_ty tau) x :=
fun n => match n with
| Some fin_n => (ap) (ren_ty (shift)) (Eq fin_n)
| None => eq_refl
end.
Definition upExt_list_ty_ty { p : nat } { m : nat } { nty : nat } (sigma : (fin) (m) -> ty (nty)) (tau : (fin) (m) -> ty (nty)) (Eq : forall x, sigma x = tau x) : forall x, (upList_ty_ty p sigma) x = (upList_ty_ty p tau) x :=
fun n => scons_p_congr (fun n => eq_refl) (fun n => (ap) (ren_ty (shift_p p)) (Eq n)).
Fixpoint ext_ty { mty : nat } { nty : nat } (sigmaty : (fin) (mty) -> ty (nty)) (tauty : (fin) (mty) -> ty (nty)) (Eqty : forall x, sigmaty x = tauty x) (s : ty (mty)) : subst_ty sigmaty s = subst_ty tauty s :=
match s with
| var_ty (_) s => Eqty s
| top (_) => congr_top
| arr (_) s0 => congr_arr ((ext_ty sigmaty tauty Eqty) s0)
| all (_) s0 => congr_all ((ext_ty (up_ty_ty sigmaty) (up_ty_ty tauty) (upExt_ty_ty (_) (_) Eqty)) s0)
| recty (_) s0 => congr_recty ((list_ext (prod_ext (fun x => (eq_refl) x) (ext_ty sigmaty tauty Eqty))) s0)
end.
Definition up_ren_ren_ty_ty { k : nat } { l : nat } { m : nat } (xi : (fin) (k) -> (fin) (l)) (tau : (fin) (l) -> (fin) (m)) (theta : (fin) (k) -> (fin) (m)) (Eq : forall x, ((funcomp) tau xi) x = theta x) : forall x, ((funcomp) (upRen_ty_ty tau) (upRen_ty_ty xi)) x = (upRen_ty_ty theta) x :=
up_ren_ren xi tau theta Eq.
Definition up_ren_ren_list_ty_ty { p : nat } { k : nat } { l : nat } { m : nat } (xi : (fin) (k) -> (fin) (l)) (tau : (fin) (l) -> (fin) (m)) (theta : (fin) (k) -> (fin) (m)) (Eq : forall x, ((funcomp) tau xi) x = theta x) : forall x, ((funcomp) (upRenList_ty_ty p tau) (upRenList_ty_ty p xi)) x = (upRenList_ty_ty p theta) x :=
up_ren_ren_p Eq.
Fixpoint compRenRen_ty { kty : nat } { lty : nat } { mty : nat } (xity : (fin) (mty) -> (fin) (kty)) (zetaty : (fin) (kty) -> (fin) (lty)) (rhoty : (fin) (mty) -> (fin) (lty)) (Eqty : forall x, ((funcomp) zetaty xity) x = rhoty x) (s : ty (mty)) : ren_ty zetaty (ren_ty xity s) = ren_ty rhoty s :=
match s with
| var_ty (_) s => (ap) (var_ty (lty)) (Eqty s)
| top (_) => congr_top
| arr (_) s0 => congr_arr ((compRenRen_ty xity zetaty rhoty Eqty) s0)
| all (_) s0 => congr_all ((compRenRen_ty (upRen_ty_ty xity) (upRen_ty_ty zetaty) (upRen_ty_ty rhoty) (up_ren_ren (_) (_) (_) Eqty)) s0)
| recty (_) s0 => congr_recty ((list_comp (prod_comp (fun x => (eq_refl) x) (compRenRen_ty xity zetaty rhoty Eqty))) s0)
end.
Definition up_ren_subst_ty_ty { k : nat } { l : nat } { mty : nat } (xi : (fin) (k) -> (fin) (l)) (tau : (fin) (l) -> ty (mty)) (theta : (fin) (k) -> ty (mty)) (Eq : forall x, ((funcomp) tau xi) x = theta x) : forall x, ((funcomp) (up_ty_ty tau) (upRen_ty_ty xi)) x = (up_ty_ty theta) x :=
fun n => match n with
| Some fin_n => (ap) (ren_ty (shift)) (Eq fin_n)
| None => eq_refl
end.
Definition up_ren_subst_list_ty_ty { p : nat } { k : nat } { l : nat } { mty : nat } (xi : (fin) (k) -> (fin) (l)) (tau : (fin) (l) -> ty (mty)) (theta : (fin) (k) -> ty (mty)) (Eq : forall x, ((funcomp) tau xi) x = theta x) : forall x, ((funcomp) (upList_ty_ty p tau) (upRenList_ty_ty p xi)) x = (upList_ty_ty p theta) x :=
fun n => (eq_trans) (scons_p_comp' (_) (_) (_) n) (scons_p_congr (fun z => scons_p_head' (_) (_) z) (fun z => (eq_trans) (scons_p_tail' (_) (_) (xi z)) ((ap) (ren_ty (shift_p p)) (Eq z)))).
Fixpoint compRenSubst_ty { kty : nat } { lty : nat } { mty : nat } (xity : (fin) (mty) -> (fin) (kty)) (tauty : (fin) (kty) -> ty (lty)) (thetaty : (fin) (mty) -> ty (lty)) (Eqty : forall x, ((funcomp) tauty xity) x = thetaty x) (s : ty (mty)) : subst_ty tauty (ren_ty xity s) = subst_ty thetaty s :=
match s with
| var_ty (_) s => Eqty s
| top (_) => congr_top
| arr (_) s0 => congr_arr ((compRenSubst_ty xity tauty thetaty Eqty) s0)
| all (_) s0 => congr_all ((compRenSubst_ty (upRen_ty_ty xity) (up_ty_ty tauty) (up_ty_ty thetaty) (up_ren_subst_ty_ty (_) (_) (_) Eqty)) s0)
| recty (_) s0 => congr_recty ((list_comp (prod_comp (fun x => (eq_refl) x) (compRenSubst_ty xity tauty thetaty Eqty))) s0)
end.
Definition up_subst_ren_ty_ty { k : nat } { lty : nat } { mty : nat } (sigma : (fin) (k) -> ty (lty)) (zetaty : (fin) (lty) -> (fin) (mty)) (theta : (fin) (k) -> ty (mty)) (Eq : forall x, ((funcomp) (ren_ty zetaty) sigma) x = theta x) : forall x, ((funcomp) (ren_ty (upRen_ty_ty zetaty)) (up_ty_ty sigma)) x = (up_ty_ty theta) x :=
fun n => match n with
| Some fin_n => (eq_trans) (compRenRen_ty (shift) (upRen_ty_ty zetaty) ((funcomp) (shift) zetaty) (fun x => eq_refl) (sigma fin_n)) ((eq_trans) ((eq_sym) (compRenRen_ty zetaty (shift) ((funcomp) (shift) zetaty) (fun x => eq_refl) (sigma fin_n))) ((ap) (ren_ty (shift)) (Eq fin_n)))
| None => eq_refl
end.
Definition up_subst_ren_list_ty_ty { p : nat } { k : nat } { lty : nat } { mty : nat } (sigma : (fin) (k) -> ty (lty)) (zetaty : (fin) (lty) -> (fin) (mty)) (theta : (fin) (k) -> ty (mty)) (Eq : forall x, ((funcomp) (ren_ty zetaty) sigma) x = theta x) : forall x, ((funcomp) (ren_ty (upRenList_ty_ty p zetaty)) (upList_ty_ty p sigma)) x = (upList_ty_ty p theta) x :=
fun n => (eq_trans) (scons_p_comp' (_) (_) (_) n) (scons_p_congr (fun x => (ap) (var_ty (p+ mty)) (scons_p_head' (_) (_) x)) (fun n => (eq_trans) (compRenRen_ty (shift_p p) (upRenList_ty_ty p zetaty) ((funcomp) (shift_p p) zetaty) (fun x => scons_p_tail' (_) (_) x) (sigma n)) ((eq_trans) ((eq_sym) (compRenRen_ty zetaty (shift_p p) ((funcomp) (shift_p p) zetaty) (fun x => eq_refl) (sigma n))) ((ap) (ren_ty (shift_p p)) (Eq n))))).
Fixpoint compSubstRen_ty { kty : nat } { lty : nat } { mty : nat } (sigmaty : (fin) (mty) -> ty (kty)) (zetaty : (fin) (kty) -> (fin) (lty)) (thetaty : (fin) (mty) -> ty (lty)) (Eqty : forall x, ((funcomp) (ren_ty zetaty) sigmaty) x = thetaty x) (s : ty (mty)) : ren_ty zetaty (subst_ty sigmaty s) = subst_ty thetaty s :=
match s with
| var_ty (_) s => Eqty s
| top (_) => congr_top
| arr (_) s0 => congr_arr ((compSubstRen_ty sigmaty zetaty thetaty Eqty) s0)
| all (_) s0 => congr_all ((compSubstRen_ty (up_ty_ty sigmaty) (upRen_ty_ty zetaty) (up_ty_ty thetaty) (up_subst_ren_ty_ty (_) (_) (_) Eqty)) s0)
| recty (_) s0 => congr_recty ((list_comp (prod_comp (fun x => (eq_refl) x) (compSubstRen_ty sigmaty zetaty thetaty Eqty))) s0)
end.
Definition up_subst_subst_ty_ty { k : nat } { lty : nat } { mty : nat } (sigma : (fin) (k) -> ty (lty)) (tauty : (fin) (lty) -> ty (mty)) (theta : (fin) (k) -> ty (mty)) (Eq : forall x, ((funcomp) (subst_ty tauty) sigma) x = theta x) : forall x, ((funcomp) (subst_ty (up_ty_ty tauty)) (up_ty_ty sigma)) x = (up_ty_ty theta) x :=
fun n => match n with
| Some fin_n => (eq_trans) (compRenSubst_ty (shift) (up_ty_ty tauty) ((funcomp) (up_ty_ty tauty) (shift)) (fun x => eq_refl) (sigma fin_n)) ((eq_trans) ((eq_sym) (compSubstRen_ty tauty (shift) ((funcomp) (ren_ty (shift)) tauty) (fun x => eq_refl) (sigma fin_n))) ((ap) (ren_ty (shift)) (Eq fin_n)))
| None => eq_refl
end.
Definition up_subst_subst_list_ty_ty { p : nat } { k : nat } { lty : nat } { mty : nat } (sigma : (fin) (k) -> ty (lty)) (tauty : (fin) (lty) -> ty (mty)) (theta : (fin) (k) -> ty (mty)) (Eq : forall x, ((funcomp) (subst_ty tauty) sigma) x = theta x) : forall x, ((funcomp) (subst_ty (upList_ty_ty p tauty)) (upList_ty_ty p sigma)) x = (upList_ty_ty p theta) x :=
fun n => (eq_trans) (scons_p_comp' ((funcomp) (var_ty (p+ lty)) (zero_p p)) (_) (_) n) (scons_p_congr (fun x => scons_p_head' (_) (fun z => ren_ty (shift_p p) (_)) x) (fun n => (eq_trans) (compRenSubst_ty (shift_p p) (upList_ty_ty p tauty) ((funcomp) (upList_ty_ty p tauty) (shift_p p)) (fun x => eq_refl) (sigma n)) ((eq_trans) ((eq_sym) (compSubstRen_ty tauty (shift_p p) (_) (fun x => (eq_sym) (scons_p_tail' (_) (_) x)) (sigma n))) ((ap) (ren_ty (shift_p p)) (Eq n))))).
Fixpoint compSubstSubst_ty { kty : nat } { lty : nat } { mty : nat } (sigmaty : (fin) (mty) -> ty (kty)) (tauty : (fin) (kty) -> ty (lty)) (thetaty : (fin) (mty) -> ty (lty)) (Eqty : forall x, ((funcomp) (subst_ty tauty) sigmaty) x = thetaty x) (s : ty (mty)) : subst_ty tauty (subst_ty sigmaty s) = subst_ty thetaty s :=
match s with
| var_ty (_) s => Eqty s
| top (_) => congr_top
| arr (_) s0 => congr_arr ((compSubstSubst_ty sigmaty tauty thetaty Eqty) s0)
| all (_) s0 => congr_all ((compSubstSubst_ty (up_ty_ty sigmaty) (up_ty_ty tauty) (up_ty_ty thetaty) (up_subst_subst_ty_ty (_) (_) (_) Eqty)) s0)
| recty (_) s0 => congr_recty ((list_comp (prod_comp (fun x => (eq_refl) x) (compSubstSubst_ty sigmaty tauty thetaty Eqty))) s0)
end.
Definition rinstInst_up_ty_ty { m : nat } { nty : nat } (xi : (fin) (m) -> (fin) (nty)) (sigma : (fin) (m) -> ty (nty)) (Eq : forall x, ((funcomp) (var_ty (nty)) xi) x = sigma x) : forall x, ((funcomp) (var_ty ((S) nty)) (upRen_ty_ty xi)) x = (up_ty_ty sigma) x :=
fun n => match n with
| Some fin_n => (ap) (ren_ty (shift)) (Eq fin_n)
| None => eq_refl
end.
Definition rinstInst_up_list_ty_ty { p : nat } { m : nat } { nty : nat } (xi : (fin) (m) -> (fin) (nty)) (sigma : (fin) (m) -> ty (nty)) (Eq : forall x, ((funcomp) (var_ty (nty)) xi) x = sigma x) : forall x, ((funcomp) (var_ty (p+ nty)) (upRenList_ty_ty p xi)) x = (upList_ty_ty p sigma) x :=
fun n => (eq_trans) (scons_p_comp' (_) (_) (var_ty (p+ nty)) n) (scons_p_congr (fun z => eq_refl) (fun n => (ap) (ren_ty (shift_p p)) (Eq n))).
Fixpoint rinst_inst_ty { mty : nat } { nty : nat } (xity : (fin) (mty) -> (fin) (nty)) (sigmaty : (fin) (mty) -> ty (nty)) (Eqty : forall x, ((funcomp) (var_ty (nty)) xity) x = sigmaty x) (s : ty (mty)) : ren_ty xity s = subst_ty sigmaty s :=
match s with
| var_ty (_) s => Eqty s
| top (_) => congr_top
| arr (_) s0 => congr_arr ((rinst_inst_ty xity sigmaty Eqty) s0)
| all (_) s0 => congr_all ((rinst_inst_ty (upRen_ty_ty xity) (up_ty_ty sigmaty) (rinstInst_up_ty_ty (_) (_) Eqty)) s0)
| recty (_) s0 => congr_recty ((list_ext (prod_ext (fun x => (eq_refl) x) (rinst_inst_ty xity sigmaty Eqty))) s0)
end.
Lemma rinstInst_ty { mty : nat } { nty : nat } (xity : (fin) (mty) -> (fin) (nty)) : ren_ty xity = subst_ty ((funcomp) (var_ty (nty)) xity) .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun x => rinst_inst_ty xity (_) (fun n => eq_refl) x)). Qed.
Lemma instId_ty { mty : nat } : subst_ty (var_ty (mty)) = id .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun x => idSubst_ty (var_ty (mty)) (fun n => eq_refl) ((id) x))). Qed.
Lemma rinstId_ty { mty : nat } : @ren_ty (mty) (mty) (id) = id .
Proof. exact ((eq_trans) (rinstInst_ty ((id) (_))) instId_ty). Qed.
Lemma varL_ty { mty : nat } { nty : nat } (sigmaty : (fin) (mty) -> ty (nty)) : (funcomp) (subst_ty sigmaty) (var_ty (mty)) = sigmaty .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun x => eq_refl)). Qed.
Lemma varLRen_ty { mty : nat } { nty : nat } (xity : (fin) (mty) -> (fin) (nty)) : (funcomp) (ren_ty xity) (var_ty (mty)) = (funcomp) (var_ty (nty)) xity .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun x => eq_refl)). Qed.
Lemma compComp_ty { kty : nat } { lty : nat } { mty : nat } (sigmaty : (fin) (mty) -> ty (kty)) (tauty : (fin) (kty) -> ty (lty)) (s : ty (mty)) : subst_ty tauty (subst_ty sigmaty s) = subst_ty ((funcomp) (subst_ty tauty) sigmaty) s .
Proof. exact (compSubstSubst_ty sigmaty tauty (_) (fun n => eq_refl) s). Qed.
Lemma compComp'_ty { kty : nat } { lty : nat } { mty : nat } (sigmaty : (fin) (mty) -> ty (kty)) (tauty : (fin) (kty) -> ty (lty)) : (funcomp) (subst_ty tauty) (subst_ty sigmaty) = subst_ty ((funcomp) (subst_ty tauty) sigmaty) .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun n => compComp_ty sigmaty tauty n)). Qed.
Lemma compRen_ty { kty : nat } { lty : nat } { mty : nat } (sigmaty : (fin) (mty) -> ty (kty)) (zetaty : (fin) (kty) -> (fin) (lty)) (s : ty (mty)) : ren_ty zetaty (subst_ty sigmaty s) = subst_ty ((funcomp) (ren_ty zetaty) sigmaty) s .
Proof. exact (compSubstRen_ty sigmaty zetaty (_) (fun n => eq_refl) s). Qed.
Lemma compRen'_ty { kty : nat } { lty : nat } { mty : nat } (sigmaty : (fin) (mty) -> ty (kty)) (zetaty : (fin) (kty) -> (fin) (lty)) : (funcomp) (ren_ty zetaty) (subst_ty sigmaty) = subst_ty ((funcomp) (ren_ty zetaty) sigmaty) .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun n => compRen_ty sigmaty zetaty n)). Qed.
Lemma renComp_ty { kty : nat } { lty : nat } { mty : nat } (xity : (fin) (mty) -> (fin) (kty)) (tauty : (fin) (kty) -> ty (lty)) (s : ty (mty)) : subst_ty tauty (ren_ty xity s) = subst_ty ((funcomp) tauty xity) s .
Proof. exact (compRenSubst_ty xity tauty (_) (fun n => eq_refl) s). Qed.
Lemma renComp'_ty { kty : nat } { lty : nat } { mty : nat } (xity : (fin) (mty) -> (fin) (kty)) (tauty : (fin) (kty) -> ty (lty)) : (funcomp) (subst_ty tauty) (ren_ty xity) = subst_ty ((funcomp) tauty xity) .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun n => renComp_ty xity tauty n)). Qed.
Lemma renRen_ty { kty : nat } { lty : nat } { mty : nat } (xity : (fin) (mty) -> (fin) (kty)) (zetaty : (fin) (kty) -> (fin) (lty)) (s : ty (mty)) : ren_ty zetaty (ren_ty xity s) = ren_ty ((funcomp) zetaty xity) s .
Proof. exact (compRenRen_ty xity zetaty (_) (fun n => eq_refl) s). Qed.
Lemma renRen'_ty { kty : nat } { lty : nat } { mty : nat } (xity : (fin) (mty) -> (fin) (kty)) (zetaty : (fin) (kty) -> (fin) (lty)) : (funcomp) (ren_ty zetaty) (ren_ty xity) = ren_ty ((funcomp) zetaty xity) .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun n => renRen_ty xity zetaty n)). Qed.
End ty.
Arguments var_ty {nty}.
Arguments top {nty}.
Arguments arr {nty}.
Arguments all {nty}.
Arguments recty {nty}.
Global Instance Subst_ty { mty : nat } { nty : nat } : Subst1 ((fin) (mty) -> ty (nty)) (ty (mty)) (ty (nty)) := @subst_ty (mty) (nty) .
Global Instance Ren_ty { mty : nat } { nty : nat } : Ren1 ((fin) (mty) -> (fin) (nty)) (ty (mty)) (ty (nty)) := @ren_ty (mty) (nty) .
Global Instance VarInstance_ty { mty : nat } : Var ((fin) (mty)) (ty (mty)) := @var_ty (mty) .
Notation "x '__ty'" := (var_ty x) (at level 5, format "x __ty") : subst_scope.
Notation "x '__ty'" := (@ids (_) (_) VarInstance_ty x) (at level 5, only printing, format "x __ty") : subst_scope.
Notation "'var'" := (var_ty) (only printing, at level 1) : subst_scope.
Class Up_ty X Y := up_ty : X -> Y.
Notation "↑__ty" := (up_ty) (only printing) : subst_scope.
Notation "↑__ty" := (up_ty_ty) (only printing) : subst_scope.
Global Instance Up_ty_ty { m : nat } { nty : nat } : Up_ty (_) (_) := @up_ty_ty (m) (nty) .
Notation "s [ sigmaty ]" := (subst_ty sigmaty s) (at level 7, left associativity, only printing) : subst_scope.
Notation "[ sigmaty ]" := (subst_ty sigmaty) (at level 1, left associativity, only printing) : fscope.
Notation "s ⟨ xity ⟩" := (ren_ty xity s) (at level 7, left associativity, only printing) : subst_scope.
Notation "⟨ xity ⟩" := (ren_ty xity) (at level 1, left associativity, only printing) : fscope.
Ltac auto_unfold := repeat unfold subst1, subst2, Subst1, Subst2, ids, ren1, ren2, Ren1, Ren2, Subst_ty, Ren_ty, VarInstance_ty.
Tactic Notation "auto_unfold" "in" "*" := repeat unfold subst1, subst2, Subst1, Subst2, ids, ren1, ren2, Ren1, Ren2, Subst_ty, Ren_ty, VarInstance_ty in *.
Ltac asimpl' := repeat first [progress rewrite ?instId_ty| progress rewrite ?compComp_ty| progress rewrite ?compComp'_ty| progress rewrite ?rinstId_ty| progress rewrite ?compRen_ty| progress rewrite ?compRen'_ty| progress rewrite ?renComp_ty| progress rewrite ?renComp'_ty| progress rewrite ?renRen_ty| progress rewrite ?renRen'_ty| progress rewrite ?varL_ty| progress rewrite ?varLRen_ty| progress (unfold up_ren, upRen_ty_ty, upRenList_ty_ty, up_ty_ty, upList_ty_ty)| progress (cbn [subst_ty ren_ty])| fsimpl].
Ltac asimpl := repeat try unfold_funcomp; auto_unfold in *; asimpl'; repeat try unfold_funcomp.
Tactic Notation "asimpl" "in" hyp(J) := revert J; asimpl; intros J.
Tactic Notation "auto_case" := auto_case (asimpl; cbn; eauto).
Tactic Notation "asimpl" "in" "*" := auto_unfold in *; repeat first [progress rewrite ?instId_ty in *| progress rewrite ?compComp_ty in *| progress rewrite ?compComp'_ty in *| progress rewrite ?rinstId_ty in *| progress rewrite ?compRen_ty in *| progress rewrite ?compRen'_ty in *| progress rewrite ?renComp_ty in *| progress rewrite ?renComp'_ty in *| progress rewrite ?renRen_ty in *| progress rewrite ?renRen'_ty in *| progress rewrite ?varL_ty in *| progress rewrite ?varLRen_ty in *| progress (unfold up_ren, upRen_ty_ty, upRenList_ty_ty, up_ty_ty, upList_ty_ty in *)| progress (cbn [subst_ty ren_ty] in *)| fsimpl in *].
Ltac substify := auto_unfold; try repeat (erewrite rinstInst_ty).
Ltac renamify := auto_unfold; try repeat (erewrite <- rinstInst_ty).